To determine the zeros of the function [tex]\( g(x) = (x^2 - 1)(x^2 - 2x + 1) \)[/tex], let's proceed step by step by examining each factor separately.
1. First factor: [tex]\( x^2 - 1 \)[/tex]:
[tex]\[
x^2 - 1 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x^2 = 1
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm 1
\][/tex]
So, the zeros of the first factor are [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
2. Second factor: [tex]\( x^2 - 2x + 1 \)[/tex]:
[tex]\[
x^2 - 2x + 1 = 0
\][/tex]
Notice that this can be factored as a perfect square:
[tex]\[
(x - 1)^2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 1 = 0 \implies x = 1
\][/tex]
So, the zero of the second factor is [tex]\( x = 1 \)[/tex].
3. Combining the zeros:
From the first factor, we have the zeros [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex]. From the second factor, we have the zero [tex]\( x = 1 \)[/tex].
Combining these zeros, the complete set of zeros is:
[tex]\[
\{-1, 1, 1\}
\][/tex]
4. Identifying distinct zeros:
The distinct zeros among [tex]\(-1, 1, 1\)[/tex] are simply [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
5. Counting the distinct zeros:
There are 2 distinct real zeros: [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
Hence, the correct statement is:
- The function has two distinct real zeros.
